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Theorem tgbtwntriv2 26272
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgbtwntriv2 (𝜑𝐵 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprl 769 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥))
2 tkgeom.p . . . . . 6 𝑃 = (Base‘𝐺)
3 tkgeom.d . . . . . 6 = (dist‘𝐺)
4 tkgeom.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (𝜑𝐵𝑃)
87ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵𝑃)
9 simplr 767 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝑥𝑃)
10 simpr 487 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → (𝐵 𝑥) = (𝐵 𝐵))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 26248 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵 = 𝑥)
1211adantrl 714 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 = 𝑥)
1312oveq2d 7171 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
141, 13eleqtrrd 2916 . 2 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15 tgbtwntriv2.1 . . 3 (𝜑𝐴𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 26249 . 2 (𝜑 → ∃𝑥𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵)))
1714, 16r19.29a 3289 1 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  cfv 6354  (class class class)co 7155  Basecbs 16482  distcds 16573  TarskiGcstrkg 26215  Itvcitv 26221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-trkgc 26233  df-trkgcb 26235  df-trkg 26238
This theorem is referenced by:  tgbtwncom  26273  tgbtwntriv1  26276  tgcolg  26339  legid  26372  hlid  26394  lnhl  26400  tglinerflx2  26419  mirreu3  26439  mirconn  26463  symquadlem  26474  outpasch  26540  hlpasch  26541
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